The purpose of this review is to apply geometric frameworks in identification problems. In contrast to the qualitative theory of dynamical systems (DSQT), the chaos and catastrophes, researches on the application of geometric frameworks have not been performed in identification problems. The direct transfer of DSQT ideas is inefficient through the peculiarities of identification systems. In this paper, the attempt is made based on the latest researches in this field. A methodology for the synthesis of geometric frameworks (GF) is propose d , which reflects features of nonlinear systems. Methods based on GF analysis are developed for the decision-making on properties and structure of nonlinear systems. The problem solution of structural identifiability is obtain ed for nonlinear systems under uncertainty.
Saha, B., Goebel, K., Poll, S. and Christopherson, J. (2009) Prognostics Methods for Battery Health Monitoring Using a Bayesian Framework. IEEE Transactions on Instrumentation and Measurement, 58, 291-296. https://doi.org/10.1109/TIM.2008.2005965
Pillonetto, G., Dinuzzo, F., Chen, T., De Nicolao, G. and Ljung, L. (2014) Kernel Methods in System Identification, Machine Learning and Function Estimation: A Survey. Automatica, 50, 657-682. https://doi.org/10.1016/j.automatica.2014.01.001
Toth, R., Sanandaij, B.M., Poolla K. and Vincent, T.L. (2012) Compressive System Identification in the Linear Time-Invariant Framework. 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, 12-15 December 2011, 783-790.
Pintelon, R. and Schoukens, J. (2012) System Identification. A Frequency Domain Approach. 2nd Edition, John Wiley & Sons, Inc., Hoboken. https://doi.org/10.1002/9781118287422
Varna, A.L., Swaminathan, A. and Wu, M. (2008) A Decision Theoretic Framework for Analyzing Binary Hash-based Content Identification. Proceedings of the 8th ACM Workshop on Digital Rights Management, Alexandria, October 2008, 67-76. https://doi.org/10.1145/1456520.1456532
Lauer, F., Bloch, G. and Vidal, R. (2011) A Continuous Optimization Framework for Hybrid System Identification. Automatica, 47, 608-613. https://doi.org/10.1016/j.automatica.2011.01.020
Iqbal, A., Ullah, I., Saeed, M.A. and Husen, A. (2019) A Classification Framework to Detect DoS Attacks. International Journal of Computer Network and Information Security, 11, 40-47. https://doi.org/10.5815/ijcnis.2019.09.05
Abdallah, H.M., Taha, A. and Selim, M.M. (2019) Cloud-Based Framework for Efficient Storage of Unstructured Patient Health Records. International Journal of Computer Network and Information Security, 11, 10-21. https://doi.org/10.5815/ijcnis.2019.06.02
Chen, L. and Narendra, K.S. (2004) Identification and Control of a Nonlinear Discrete-Time System Based on its Linearization: A Unified Framework. IEEE Transactions on Neural Networks, 15, 663-673. https://doi.org/10.1109/TNN.2004.826206
Kurtoglu, T. and Tumer, I.Y. (2008) A Graph-Based Fault Identification and Propagation Framework for Functional Design of Complex Systems. Journal of Mechanical Design, 130, Article ID: 051401. https://doi.org/10.1115/1.2885181
Papadopoulos, P.N., Guo, T. and Milanović, J.V. (2018) Probabilistic Framework for Online Identification of Dynamic Behavior of Power Systems with Renewable Generation. IEEE Transactions on Power Systems, 33, 45-54. https://doi.org/10.1109/TPWRS.2017.2688446
Roettgen, D., Allen, M.S., Kammer, D. and Mayes, R.L. (2017) Substructuring of a Nonlinear Beam Using a Modal Iwan Framework, Part I: Nonlinear Modal Model Identification. In Allen, M., Mayes, R. and Rixen, D., Eds., Dynamics of Coupled Structures, Vol. 4, Springer, Cham, 165-178. https://doi.org/10.1007/978-3-319-54930-9_15
Carino, J.A., Delgado-Prieto, M., Iglesias, J.A. and Sanchis, A. (2018) Fault Detection and Identification Methodology under an Incremental Learning Framework Applied to Industrial Machinery. IEEE Access, 6, 49755-49766. https://doi.org/10.1109/ACCESS.2018.2868430
The, C.Y., Kerk, Y.W., Tay, K.M. and Lim, C.P. (2018) On Modeling of Data-Driven Monotone Zero-Order TSK Fuzzy Inference Systems Using a System Identification Framework. IEEE Transactions on Fuzzy Systems, 26, 3860-3874. https://doi.org/10.1109/TFUZZ.2018.2851258
Nagarajaiah, S. (2017) Sparse and Low-Rank Methods in Structural System Identification and Monitoring. Procedia Engineering, 199, 62-69. https://doi.org/10.1016/j.proeng.2017.09.153
Wiggins, S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer Science & Business Media, Berlin.
Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V. and Chua, L.O. (2001) Methods of Qualitative Theory in Nonlinear Dynamics (Part II). World Scientific, Hackensack. https://doi.org/10.1142/4221
Michel, A., Wang, K. and Hu, B. (2001) Qualitative Theory of Dynamical Systems. CRC Press, Boca Raton. https://doi.org/10.1201/9780203908297
Pershin, Y.V. and Slipko, V.A. (2018) Dynamical Attractors of Memristors and Their Networks. Europhysics Letters, 125, 20002. https://doi.org/10.1209/0295-5075/125/20002
Lu, J. and Zhang, S. (2001) Controlling Chen’s Chaotic Attractor Using Backstepping Design Based on Parameters Identification. Physics Letters A, 286, 148-152. https://doi.org/10.1016/S0375-9601(01)00383-8
Li, C., Min, F. and Li, C. (2018) Multiple Coexisting Attractors of the Serial-Parallel Memristor-Based Chaotic System and Its Adaptive Generalized Synchronization. Nonlinear Dynamics, 94, 2785-2806. https://doi.org/10.1007/s11071-018-4524-3
Karabutov, N. (2017) Structural Methods of Design Identification Systems. In: Uvarova, L., Nadykto, A.B. and Latyshev, A.V., Eds., Nonlinearity: Problems, Solutions and Applications, Vol. 1, Nova Science Publishers Inc., New York, 233-274.
Karabutov, N. (2018) Frameworks in Identification Problems: Design and Analysis. URSS/Lenand, Moscow.
Achho, L. (2013) Hysteresis Modeling and Synchronization of a Class of RC-OTA Hysteretic-Jounce-Chaotic Oscillators. Universal Journal of Applied Mathematics, 1, 82-85. https://doi.org/10.13189/ujam.2013.010207
Gao, H., Jézéquel, L., Cabrol, E. and Vitry, B. (2019) Characterization of a Bouc-Wen Model-Based Damper Model for Automobile Comfort Simulation. Surveillance, Vishno and AVE Conferences, Lyon, Jul 2019, hal-02188563.
Karabutov, N. (2015) Structural Methods of Estimation Lyapunov Exponents Linear Dynamic System. International Journal of Intelligent Systems and Applications, 7, 1-11. https://doi.org/10.5815/ijisa.2015.10.01
Karabutov, N. (2018) About Structural Identifiability of Nonlinear Dynamic Systems under Uncertainty. International Journal of Intelligent Systems and Applications, 18, 51-61. http://doi.org/10.5815/ijisa.2020.01.02
Bylov, F., Vinograd, R.E., Grobman, D.M. and Nemytskii, V.V. (1966) Theory of Lyapunov Exponents and Its Application to Problems of Stability. Nauka, Moscow.
Karabutov, N. (2018) About Lyapunov Exponents Identification for Systems with Periodic Coefficients. International Journal of Intelligent Systems and Applications, 10, 1-10. https://doi.org/10.5815/ijisa.2018.11.01
Bodunov, N.A. (2012.) Introduction to the Theory of Local Parametrical Identifiability. Differential Equations and Management Processes.
Saccomani, M.P. (2013) Structural vs Practical Identiability in System Biology. 2013 International Work-Conference on Bio Informatics and Biomedical Engineering, Granada, 18-20 March 2013, 305-313.
Chis, O.-T., Banga, J.R. and Balsa-Canto, E. (2011) Structural Identifiability of Systems Biology Models: A Critical Comparison of Methods. PLoS ONE, 6, e27755. https://doi.org/10.1371/journal.pone.0027755
Stigter, J.D. and Peeters, R.L.M. (2007) On a Geometric Approach to the Structural Identifiability Problem and Its Application in a Water Quality Case Study. 2007 European Control Conference, Kos, 2-5 July 2007, 3450-3456. https://doi.org/10.23919/ECC.2007.7068560
Krasovsky, A.A. (1987) Reference Book on Automatic Control Theory. Nauka, Moscow.
Takens, F. (1980) Detecting Strange Attractors in Turbulence. In: Rand, D.A. and Young, L.-S., Eds., Dynamical Systems and Turbulence, Warwick 1980, Vol. 898, Springer-Verlag, Berlin, Heidelberg, 366-381. https://doi.org/10.1007/BFb0091924
Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A. (1985) Determining Lyapunov Exponents from a Time Series. Physica D: Nonlinear Phenomena, 16, 285-301. https://doi.org/10.1016/0167-2789(85)90011-9
Rosenstein, M.T., Collins, J.J. and De Luca, C.J. (1993) A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets Source. Physica D: Nonlinear Phenomena, 65, 117-134. https://doi.org/10.1016/0167-2789(93)90009-P
Anishchenko, V.S., Astakhov, V., Neiman, A. Vadivasova, T. and Schimansky-Geier, L. (2007) Nonlinear Dynamics of Chaotic and Stochastic Systems. Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments. 2nd Edition, Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38168-6
Malinetskiy, G.G. and Potapov, A.B. (2000) Modern Problems of Nonlinear Dynamics. Editorial URSS, Moscow.
Pecora, L.M., Moniz, L., Nichols, J. and Carroll, T.L. (2007) A Unified Approach to Attractor Reconstruction. Chaos, 17, Article ID: 013110. https://doi.org/10.1063/1.2430294
Bradley, E. and Kantz, H. (2015) Nonlinear Time-Series Analysis Revisited. Chaos, 25, Article ID: 097610. https://doi.org/10.1063/1.4917289
Liebert, W., Pawelzik, K. and Schuster, H.G. (1991) Optimal Embedding of Chaotic Attractors from Topological Considerations. Europhysics, Letter, 14, Article No. 521. https://doi.org/10.1209/0295-5075/14/6/004
Buzug, T. and Pflster, G. (1992) Optimal Delay Time and Embedding Dimension for Delay-Time Coordinates by Analysis of the Global Static and Local Dynamical Behaviour of Strange Attractors. Physical Review A, 45, Article No. 7073. https://doi.org/10.1103/PhysRevA.45.7073
Deshpande, A., Chen, Q., Wang, Y., Lai, Y.-C.G. and Do, Y. (2010) Effect of Smoothing on Robust Chaos. Physical Review E, 82, Article ID: 026209. https://doi.org/10.1103/PhysRevE.82.026209
Aguirre, L.A. and Letellier, C. (2009) Modeling Nonlinear Dynamics and Chaos: A Review. Mathematical Problems in Engineering, 2009, Article ID: 238960. https://doi.org/10.1155/2009/238960
Johnston, J. (1972) Econometric Methods. 2nd Edition, McGraw-Hill Book Company, New York.
Armstrong, B. (2006) Models for the Relationship between Ambient Temperature and Daily Mortality. Epidemiology, 17, 624-631. https://doi.org/10.1097/01.ede.0000239732.50999.8f
Malinvaud, E. (1980) Statistical Methods in Econometrics. 3rd Edition, North-Holland Publishing Co., Amsterdam.
Dhrymes, P.J. (1971) Distributed Lags: Problems of Estimation and Formulation. Holden-Day, San Francisco.
Solow, R. (1960) On a Family of Lag Distributions. Econometrica, 28, 393-406. https://doi.org/10.2307/1907729